RIVARD and BLEDSOE: PARAMETER ESTIMATION FOR PELLA-TOMLINSON MODEL 



where (by assuming ft_t^\ constant over the inter- 

 val t,t + \) 



mates of m and B x- Finally, Bo is approximated by 



B 



t.t+\ i.t+i'^ 't.t + \ 



(15) 



Note that Y,j + y and /",, ^i correspond to y, and/', of 

 Equations (6) and (5). 



STEP 3. Let /z= 2, as in the Graham-Schaefer 

 model, and estimate m andB-^ by fitting the linear 

 model 





(16) 



where y = 



- Tjn—l 



t B^dt 



+ qf^, x^ = B';-\ 



Equation (16) is derived from Equations (1) and 

 (3). However, Equation (16) requires an estimate 

 of the relative growth rate dB,/Bidt, say i?,. As 

 suggested by Causton ( 1969), the mean value of 7? 

 between / and t+2 is given by 



^M.2 = (lnB^^,-lnB^)/2. 



(17) 



For the purpose of fitting Equation ( 16), quantity 

 /?,,+2 n^^y t>6 considered an estimate ofRi + i, 

 which corresponds to B, + ^. Whence. Equation 

 (16) provides estimates of/?? andSx as 



m = 



h %"] 



(l/l-n) 



7 



B = 



ct. 



d. 



1/1— ^7 



(18) 



(19) 



STEP 4. Steps 2 and 3 are repeated iteratively 

 for increasing values of q. The value of g which 

 provides the minimum residual sum of squares 



[1 (Y, - y, )2] is accepted as the appropriate start- 

 ing value for q. 



STEP 5. Step 3 is repeated iteratively for in- 

 creasing values of n, parameter q being kept con- 

 stant. The value of n which provides the minimum 

 residual sum of squares [1 (y, - Y,)^] is accepted 



as the appropriate starting value for n . In the last 

 iteration, Equations (18) and (19) provide esti- 



^0 = ^0,1 



a 



(20) 



where B, andB,, , are estimated by Equations ( 14) 

 and (15), respectively. 



Steps 1 through 5 provide a set of starting values 

 for the optimization algorithm (11). Usually the 

 starting values are near the solution and few iter- 

 ations will be needed. Of course, it would be possi- 

 ble to derive algorithms for more accurate starting 

 values, but our purpose here is to find a rough 

 estimate for each coefficient and to let the iterative 

 procedure (11) converge to the minimum. Some- 

 times, by experience or by prior information, it is 

 possible to provide starting values as satisfactory 

 as those provided by the algorithm given above. 



MONTE-CARLO SIMULATIONS 



The parameter values that we chose to generate 

 the data of Table 1 (deterministic model) were 

 recovered exactly by the estimation procedure. 

 Results of fitting 18 stochastic versions of the de- 

 terministic model are also included in Table 2. 

 Based on our simulation results, there do not ap- 

 pear to be any serious problems with bias of 

 parameter estimates. The bottom line of Table 2, 

 which gives the coefficients of variation of the 

 parameter estimates, reveals that estimates of the 

 three parameters of principal interest to the man- 

 ager have the smallest variability. Those 

 parameters are maximum sustainable yield (m, 

 C.V. = 147f ), optimal effort level(/MSY> C.V. = 67r ), 

 and yield per unit of effort at optimum effort 

 (t/ivjsY' C.V. = 9%). Our results confirm the ob- 

 servations of Fox (1971) and Pella and Tomlinson 

 (1969) on the robustness of m and /"msy with re- 

 spect to error in the measurement of the yield 

 data. From Table 2, we can also compare variance 

 estimates from Equation ( 12 » with variance of es- 

 timates for 10 replicates at o" = 0.200. Equation 

 (12) appears to give (approximately) unbiased es- 

 timates of the variance of the sampling distribu- 

 tion of G. Also, out of the 19 cases considered, the 

 true parameter value lay outside the arbitrary ±2 

 (SD) confidence interval twice form and only once 

 each forB^,/?, andB,,. Although we did not employ 

 an extensive Monte-Carlo simulation, our results 

 suggest that the normal approximation to the 

 sampling distribution of Q is an acceptable ap- 

 proximation, at least for management purposes. 



527 



